332 W
. Bossert, H. Peters Mathematical Social Sciences 40 2000 327 –339
where for every j [ M, u : A →
R is an induced utility function as introduced in Section
j j
2.
4. Multiplicative decision criteria
Keeney and Raiffa 1976 prove that if each attribute space A is utility independent
j
of A , then the cardinal utility function u can be written as a multilinear function —
2j
see Keeney and Raiffa 1976, Chaps. 5 and 6 for details. In this section, we derive the special case of a multiplicative decision criterion with a weaker utility independence
axiom and an assumption which requires the a to lead to a worst alternative even when
j
combined with some values of a other than a .
2j 2j
To exclude degenerate cases, we assume that there is an a [ A with a ± a for all
j j
¯ j [ M and ua . ua . For j [ M, define A
[ha [ A ua ± a for all
2h1, . . . , j j 2h1, . . . , j j
k k
k 5 j 1 1, . . . , mj. The utility independence condition is parallel to the one in the
¯ previous section, where a is replaced with a, and A
by A .
2h1, . . . , j j 2h1, . . . , j j
Restricted a utility independence:
¯ •
A is utility independent of A ;
1 21
¯ •
A is utility independent of haj 3 A
;
2 1
2h1,2j
• :
¯ •
A is utility independent of
haj 3 haj 3 ? ? ? 3 ha j 3 A .
m 21
1 2
m 22
2h1, . . . ,m21j
The following axiom requires a to lead to a worst alternative when combined with
j
certain values of a .
2j
Restricted zero independence:
ua 9 ua , a , for all a, a9 [ A
9
1 21
ua 9 ua, a , a
, for all a, a
9 [ A 10
1 2
2h1,2j
: ua
9 ua, . . . , a , a , a ,
for all a, a 9 [ A
11
1 m
22 m
21 m
Restricted zero independence implies that a is a worst-possible value for attribute one
1
in an absolute sense: a leads to a worst alternative not only combined with a but
1 21
combined with any value of a . The remaining restrictions imposed by the axiom are
21
weaker because they apply only to some but not to all values of the remaining attributes. Requiring restricted zero independence is close to but weaker than imposing the
so-called zero condition on attributes 1, . . . , m 2 1, as is done in Miyamoto et al.
1998. The zero condition holds for attribute j if the level a for attribute j makes the
j
decision maker indifferent between all combinations of the other attributes.
W . Bossert, H. Peters Mathematical Social Sciences 40 2000 327 –339
333
The above two axioms lead to the multiplicative decision criterion described in the following theorem.
Theorem 2. Let u satisfy restricted a utility independence and restricted zero
independence. Assume that u is normalized so that ua 5 0 and ua 5 1. Then:
ua 5
P
ua , a for all a [ A.
j 2j
j [M
Proof. By 9–11 we must have:
ua , a 5 0, for all a [ A
12
1 21
ua, a , a 5 0, for all a [ A
13
1 2
2h1,2j
: ua, . . . , a
, a , a
5 0, for all a [ A 14
1 m
22 m
21 m
¯
9
Because A is utility independent of A
, choosing a 5 a in 1 implies that
1 21
21 21
there exist functions f and g such that:
1 1
ua 5 f a 1 g a ua , a , for all a [ A
15
1 21
1 21
1 21
Letting a 5 a in 15, 12 implies f a 5 0 for all a [ A.
1 1
1 21
Now let a 5 a in 15. Noting that f a 5 0 and ua 5 1, it follows that
1 1
1 21
g a 5 ua, a for all a [ A. Therefore:
1 21
1 21
ua 5 ua, a ua , a , for all a [ A
16
1 21
1 21
¯ Because A
is utility independent of haj 3 A
, there exist functions f and g
2 1
2h1,2j 2
2
such that: ua, a
5 f a, a 1 g a, a
ua , a , for all a [ A
17
1 21
2 1
2h1,2j 2
1 2h1,2j
2 22
9
[choose a 5 a in 1]. Letting a 5 a in 17 and using 13, we obtain f a,
22 22
2 2
2 1
a 5 0 for all a [ A. Now let a 5 a in 17 to obtain g a, a
5 ua, a,
2h1,2j 2
2 2
1 2h1,2j
1 2
a for all a [ A. Hence:
2h1,2j
ua, a 5 ua, a, a
ua , a , for all a [ A
18
1 21
1 2
2h1,2j 2
2
Using 18 in 16, we obtain: ua
5 ua , a , a ua , a
ua , a ,
for all a [ A
1 2
2h1,2j 2
22 1
21
Repeated application of this argument for the attributes 3, . . . , m 2 1 yields:
ua 5 ua , a
. . . ua , a ua , a
for all a [ A. h
m 2m
2 22
1 21
Again, it follows that there exist induced cardinal utility functions u : A →
R for all
j j
j [ M such that:
334 W
. Bossert, H. Peters Mathematical Social Sciences 40 2000 327 –339
ua 5
P
u a , for all a [ A
19
j j
j [M
To conclude this section we note that the conditions in Theorem 2 are sufficient but not necessary for a multiplicative representation. For example, take m
5 2, A 5 A 5
1 2
h1 2, 1j, and ua , a 5 a a . In this case, 9 is not satisfied.
1 2
1 2
5. Efficiency
